| L(s) = 1 | + 2.56i·3-s + 1.56i·7-s − 3.56·9-s + 2·11-s + 0.561i·13-s + 1.56i·17-s − 6·19-s − 4·21-s + i·23-s − 1.43i·27-s + 2.12·29-s − 9.24·31-s + 5.12i·33-s + 0.438i·37-s − 1.43·39-s + ⋯ |
| L(s) = 1 | + 1.47i·3-s + 0.590i·7-s − 1.18·9-s + 0.603·11-s + 0.155i·13-s + 0.378i·17-s − 1.37·19-s − 0.872·21-s + 0.208i·23-s − 0.276i·27-s + 0.394·29-s − 1.66·31-s + 0.891i·33-s + 0.0720i·37-s − 0.230·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9882900182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9882900182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 - 2.56iT - 3T^{2} \) |
| 7 | \( 1 - 1.56iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.561iT - 13T^{2} \) |
| 17 | \( 1 - 1.56iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 0.438iT - 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 7.68iT - 47T^{2} \) |
| 53 | \( 1 + 0.438iT - 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 4.43iT - 67T^{2} \) |
| 71 | \( 1 - 1.87T + 71T^{2} \) |
| 73 | \( 1 + 8.56iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 - 2.24T + 89T^{2} \) |
| 97 | \( 1 - 4.87iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.362189695739923734763079104641, −8.911775451305974453188708903889, −8.236468469072239596160313732725, −7.05959873346787301227638776622, −6.12349114470830002877418782494, −5.43466003893453319520443709987, −4.49639262794295976712704068816, −3.94674267463688639654017916825, −3.02932428616861243646637013049, −1.81272631460850310706014414490,
0.32703884585831648308667288333, 1.49522588001476367536538684544, 2.32894006910848013716554516161, 3.54952314532672067303804848579, 4.50418075742600502692610010654, 5.66242896332010733462295868651, 6.47720705641153756255839716199, 7.02085720165371815436096533349, 7.61928346212063383704790655859, 8.465520580123938441425438138384
